\section{Basic Insurance Principles}
\label{sec:BasicInsurancePrinciples}

Insurance was practiced long before modern risk theory explained how risk aggregation and management through insurance works. Insurers must collect sufficient premiums to pay all their claims costs\index{Claims Costs} and non-claim related operating expenses\index{Operating Expenses}, reward policyholders or stockholders for using their capital (Profit Margin\index{Profit Margin}), and secure protection, the Risk Premium\index{Risk premium}, from their exposure to higher than expected claims costs \citep{Bowers:2e:1997, HoggLossDist1984, Salzmann1984}. Before issuing policies, insurer's have expectations about their revenues and costs, their Prospective Insurer Premium Allocation By Cost Components (See Equation~\ref{eq:PremiumAllocationByCostsProspective}):
\begin{eqnarray} \label{eq:PremiumAllocationByCostsProspective}
\textrm{Premium Revenues} & = & \textrm{Claims Costs}\nonumber  \\
	&  & + \quad \textrm{Operating Expenses} \nonumber   \\
	&  & + \quad \textrm{Profit Margin} \nonumber  \\
	&  & + \quad \textrm{Risk Premium} 
\end{eqnarray}

Insurers' premium allocations are unknown until policies expire and accounting is complete. Claims costs (Losses and Loss Adjustment Expenses) are usually insurers' highest and least certain costs. Insurers' profits, operating losses, and solvency are uncertain primarily because their claims costs are uncertain. When all claims costs have been paid, when no uncertainty exists, there are no profit margins or risk premiums, just profits or operating losses, the insurer's Retrospective Premium Allocation By Cost Components (See Equation~\ref{eq:PremiumAllocationByCostsRetrospective}):
\begin{eqnarray} \label{eq:PremiumAllocationByCostsRetrospective}
\textrm{Profit (Operating Loss)} & = & \textrm{Premium Revenues} \nonumber \\
	&  & - \quad \textrm{Claims Costs}\nonumber  \\
	&  & - \quad \textrm{Operating Expenses} 
\end{eqnarray}
